IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


I.I 


1.25 


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^  IIIIM   IIIII12 


II 


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Photographic 

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WEBSTER,  N.Y.  14580 

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CIHM/ICMH 

Microfiche 

Series. 


CIHM/ICIVIH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions 


Institut  Canadian  de  microreproductions  historiques 


1980 


Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


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D 
D 
D 
D 


Coloured  covers/ 
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Covers  damaged/ 
Couverture  endommag^e 

Covers  restored  and/or  laminated/ 
Couverture  restaurde  et/ou  pelliculde 

Cover  title  missing/ 

Le  titre  de  couverture  manque 


I      I    Coloured  maps/ 


Cartes  gdographiques  en  couleur 


□    Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 

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Blank  leaves  added  during  restoration  may 
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lors  d'une  restauratlon  apparaissent  dans  le  texte, 
mais,  lorsque  cela  6tait  possible,  ces  pages  n'ont 
pas  6t6  filmdes. 


D 


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Transparence 

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Qualitd  indgale  de  I'impression 

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Only  edition  available/ 
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I — I  Pages  damaged/ 

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I      I  Pages  detached/ 

I      I  Showthrough/ 

I      I  Quality  of  print  varies/ 

I      I  Includes  supplementary  material/ 

I — I  Only  edition  available/ 


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obtenir  la  meilleure  image  possible. 


Additional  comments:/ 
Commentaires  suppl6mentaires: 

This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  filmd  au  taux  de  reduction  indiqud  ci-dessous 

10X                            14X                            18X                            22X 

26X 

30X 

- 

. 

^/ 

12X                             16X                             20X                              24X                             28X                             32X 

The  copy  filmed  here  has  been  reproduced  thanks 
to  the  gene/osity  of: 

Library, 

Geological  Survey  of  Canada 


L'exemplaire  filmd  fut  reprodult  grSce  d  la 
g6n6rosit6  de: 

BibliotMque, 

Commission  Giologique  du  Canada 


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Les  images  suivantes  ont  6t6  reproduites  avec  le 
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de  la  netteti  de  l'exemplaire  filmd,  et  en 
conformity  avec  les  conditions  du  contrat  de 
filmage. 


Original  copies  in  printed  paper  cover';  are  filmed 
beginning  with  the  front  cover  and  ending  on 
the  last  page  with  a  printed  or  illustrated  impres- 
sion, or  the  back  cover  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


Les  exemplaires  originaux  dont  la  couverture  en 
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plat,  selon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  film6s  en  commengant  par  la 
premidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  — ^-  (meaning  "CON- 
TINUED "),  or  the  symbol  V  (meaning  "END"), 
whichever  applies. 


Un  des  symboles  suivants  apparaitra  sur  la 
dernidre  image  de  chaque  microfiche,  selon  le 
cas:  le  symbols  — ►  signifie  "A  SUIVRE".  le 
symbole  V  signifie  "FIN". 


Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Les  cartes,  planches,  tableaux,  etc..  peuvent  dtre 
film4s  d  des  taux  de  reduction  diffirents. 
Lorsque  le  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  clich6,  il  est  filmd  d  partir 
de  Tangle  sup6rieur  gauche,  de  gauche  d  droite. 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  ndcessaire.  Les  diagrammes  suivants 
illustrent  la  mdthode. 


1    "* 

1       2 

.        .          .        ■    ■ 

3 

[             t 

2 

3 

4 

5 

6 

.  *■•■ 


SMITHSONIAN  CONTRIBUTIONS  TO  KNOWLEDGE. 

■ 281 


ON    THE 


GENERAL  INTEGRALS 


PLANETARY   MOTION 


BY 


SIMOx\   NEWT 0MB, 

i-IiorKSSIlH  III-  JIATllKMAilCS     .XITKI,  STATKS  XAVY. 


I  ArCKPIKll    K(1R    IMMU.ICATIflV.    (I  C  T  II  11  K  H  ,     I  S  7  .(  .  J 


Pin  LA  nicM'M  I  a; 
CiJi.hlNs,  nUNTKU.  70q  JAYNK  8TUEET. 


ADYEHTISEMENT. 


The  following  ]\[omoir,  on  the  "  General  Integrals  of  Planetary  Motion,"  was 
submitted  to  Prof  II.  A.  Newton,  of  Yale  College,  and  Mr.  (J.  ^\.  Hill,  of  Nyuck, 
N".  Y.,  and  has  received  their  approval  for  publication  in  the  "  Smithsonian  Con- 


tributions to  Knowledge." 


WASiriNOTON,  ^^.  0., 

Deccnilx'i',  1874. 


JOSEPH  HENRY, 

Secretary  Smithsonian  Institution. 


(iii) 


•"-''•  •tiillii    Vfi'  '"iTl.J^ 


IM{K1' ACK. 


Thk  |)n'S('iit  memoir  may  he  fumsidcnd  iis.  in  pnrt,  nn  rxtciision  aiul  jj;mprnliza- 
tiou  of  two  foniicv  i)ai)ci-.s  liy  the  aiitiior:  tiic  first  Ixuiig  T/n'orir  i/rsjirr/iir/iii/iniin  </< 
III.  l.iiiir  i/ii!.s,ni./  i/iir.'i  i)  /'(iHloii.  ih'n  P/anc/cs,  piil)lisli('(l  in  J.i<umUi''n  Joiirihul,  tome 
xvi.,  1.S71  ;  and  tlu'  second,  Siir  mi.  Thro  mm:  </r  Mrriiu.ii/iir  Crh.slr,  pnhlisiied  in  tiie 
Coiiipkfi  Jiendits,  tome  Ixxv.  Notwitiistandinj,'  its  extent,  the  antlior  is  ronscions, 
Ml  his  treatment  of  tlio  snhji-rt,  of  s(-veral  gaps,  wliicli  nniy  detract  from  entire  rigor. 
lie  believes  tliat  some  of  tiiese  are  of  snch  n  natnre  tliat  tlie  reader  can  readily 
iill  them,  while  the  remaind(-r  wonld  have  led  hito  long  digressions,  and  tiins 
caused  great  delay  in  the  publication  of  th(>  paper.  To  the  fornu-r  class  belong 
(1)  tJK!  analogy  Ix'tween  the  expressions  for  tlu-  rectangular  co-ordinates  ,»•  and 
//,  wliieh  differ  only  hi  that  tlie  latter  is  comiiosed  of  jn-oducts  of  sines,  wliile  tlie 
former  is  composed  of  similar  ])rodncts  of  cosines ;  and  (•,')  the  omission  of  all 
consid(«rations  of  the  modifications  growing  out  of  tlu-  fact  tinit  in  ecpiation  (1) 
one  value  of  /i  vanishes.  To  the  latter  class  belong  the  omission  of  all  considera- 
tions r(>specting  the  convergence  of  the  series  encountered,  respecting  terms  of 
long  period,  and  respecting  the  occurrence  of  relations  among  tlie  arguments, 
such  as  that  known  to  subsist  betwi'cu  tlu;  mean  motions  of  three  of  the  sati'llites 
of  Jupiter.  These  subjects  will  naturally  come  up  for  consideration  whi'U  tiie 
process  of  actually  integrating  tlie  differential  ecpiations  of  planetary  motion  in 
the  most  general  way  is  midert.dveu.  Xo  method  for  the  actual  execution  of  this 
integration  is  given  at  present,  partly  l)ccause  tlie  papcn-  may  be  considered  com- 
plete without  it,  partly  because  the  author  has  not  succeeded  in  working  out  any 
method  satisfactory  to  himself  It  is  true  that  a  large  part  of  the  paper  is  devote<I 
to  reviewing  the  general  forms  met  with  in  a  certain  integrating  process,  but  the 
actual  execution  of  this  process,  even  for  a  single  approximation,  may  be  consid- 
ered impracticable  on  account  of  the  enormous  labor  involved  in  it.  It  is  shown, 
by  a  bird's  eye  view,  that  a  certain  olyect  is,  in  tliV  nature  of  things,  attainable ; 
but  a  practicable  way  of  actually  reaching  it  is  yc-t  to  be  point(>(l  out.  It  would  be 
extremely  agreeable  to  the  author  to  learu  that  abler  hands  than  his  were  success- 
fully working  to  effect  the  actual  solution  of  this  noble  problem  in  its  most  general 


form. 


(  V) 


CONTENTS. 


§  1.  Introduction  .  .  .  .  ,  .  .  •  .  . 

§  2.  Canonical  Transformntioii  of  tlic  Equations  of  Motion     ..... 

§  3.  Approximation  to  tlic  Kcquirt'il  Solution  Ijy  tlio  Viiriutions  of  tlio  Arbitrary  Constants 

in  ft  First  Approximate  Solution        ....... 

§  4.   Formation  of  the  Lagraiigian  ('uelDcionts  (n„  at),  and  lloduetiou  of  tlio  Equations  to  a 
Canonical  Form 

§  5.  Fundamental  Relations  between  t..o  Coellieicuts  of  tliu  time,  b„  b,,  etc.,  considered  as 
Functions  of  c,,  c,,  etc.  •.•..... 

§•  G.  Development  of  n,  iij,  and  Q;      .  .  . 

§  1.  Form  of  Second  Approximation  .  .  ,  ,  ,  ... 

§  S.   General  Tlieoreni  ••....... 

§  i).   Summary  of  Llesultd  .  .  ,  .  .  ,  . 


PAUU 
1 


11 

10 
10 
24 

20 
2H 


(  vii  ) 


ON    THE 

GENEKAL  INTEGRALS  OF  PLANETARY  MOTION. 


§  1.   Introilitcdon. 


If  wc  examine  wliiit  lius  been  done  liy  fjfeomefers  towards  developinjjf  tlie  co 
ordinate^  of  tlie  pliUiets  in  terms  of  tlie  time,  we  sliail  .see  that  tlie  iiio.st  >,'eiieral 
expressions  yet  found  are  those  f-tv  tlie  development  of  the  secular  variations  of 
the  elements  in  a  periodic  form.  It,  i^  well  known  that  if  we  negleet  (piantities 
of  tlie  third  order  with  respect  to  th*?  eccentricities  and  inclinations,  the  intei^ia- 
tion  of  the  equations  which  «iv(>  liie  secular  variations  of  those  elements,  and  of 
the  loii<,'itudes  of  the  periheh'  iid  of  llie  nodes,  h-ads  to  the  conclusion  that  tlie 
genend  expressions  of  those  elenumts  in  terms  of  the  time  are  of  the  form 


C  sin  7T 


(1) 


c  COS  71  =  S,  Xi  COS  (r/,Y  -f-  li^) 
^  sin  0  =  i\  JUi  sill  (///  -j-  y^) 
^  cos  0  =  Vj  Mi  cos  (///  -|-  y,) 

71  beiiif,'  tlie  number  of  planets-,  A',  M;,  (/,,  aiid  //,  being  functions  of  the  eccentrici- 
ties at  a  given  epoch  and  of  the  mean  distances,  whih;  ,5,  and  y^  are  angles  depend- 
ing also  on  the  positions  of  the  perihelia  and  nodes  at  a  given  epoch.  It  is  to  be 
nmiarkinl  that  one  of  the  values  of  hi  is  zero,  the  corresponding  (piantities  J/  and  y 
depending  on  the  position  of  tlie  plane  of  reference. 

The  num(>rical  values  of  these  constants  for  the  solar  system  have  been  fouiui  liy 
several  geometers.  'I'he  latest  and  most  complete  determiualions  are  thos(,'  of 
Jje  Verrier  and  of  Stockwell.' 

A\  hen  we  consider  the  terms  commonly  called  ])eriodic,  that  is,  those  wliich 
depiMid  on  the  mean  longitudes  of  the  i>lan(>ts,  we  shall  find  that  their  determina- 
tion depends  on  the  integration  of  differentials  of  the  form 

'"'^  ,i„  ('V  + // +/71' +;ri  +  ^-v  + /^?o. 

■tvher(>  we  put 

m'  the  mass  of  the  disturbing  planet. 


'  Smithsonian  CuuU'ibutioiis  to  KMowli'dj,^',  Xo.  •23-2.      \'ol.  XVJII. 
October,  1874.  j 


G  K  S  K  U  A  L    IN  T  E  G  U  A  L  S    O  V    V  L  A  \  K  T  A  R  Y    M  O  T I  O  N . 


//  ii  function  of  the  eccentricities,  inclinations,  and  mean   distances  of  the  two 
phinets,  devekjpable  in  powers  of  the  two  former  quantities. 

/,  /'  the  mean  ionyitm' vs  of  tlie  phniets. 

;t,  7i'  the  longitudes  of  their  periiielia. 

0,  0'  the  longitudes  of  their  nodes. 

(■,y, /•,  numerical  integer  coefficients, 

and  in  which  /'  +  /  +/  -f-  /  -f-  /,•'  -f  /,•  —  0. 

Tile  coefficient  7i  is  of  the  form  * 

AtA'Jyci^"''  (1  +  A,c'-\-  A.,'"  -f  etc.), 
while  the  circular  function  of  whidi  it  is  a  coefficient  may  be  put  in  the  form 

COS 

sin  '•^''^  +-'"^'  +  /'■^'  +''•'"')  cos  (/7  +  (7) 

±  c!"  (.^"^  +->"^'  +  ''■•'  +  ^•'''')  ^i»  ('''  +  '0. 
As  these  equations  have  hitlierto   been  integrated   the   different  elements  arc 
developed  in  powers  of  tlie  time,  and  w(  arc  tints  led  to  expressions  of  the  form 

(.l  +  .17  +  .l'r  +  ...,);'!;^^(;7'+//). 

But  it  is  clear,  that  we  sliall  get  more  general  expressions  if,  instead  of  using 
developnrents  in  pow(>rs  of  tlie  time,  wo  substitute  tiie  genenU  values  of  the  ele- 
ments given  by  equations  (1).  Tlio  substitution  will  be  most  readily  made  by 
reducing  the  circular  to  exponential  functions,     i  utting  in  (1)  for  brevity 


n  =  f  "N^^ 

the  equations  (1)  may  be  put  in  the  form 

ell-' =z=  v.A^A -^ 

Tn  the  preceding  differential  to  be  integrated  the  coefficient  of  *"^,  (/'/'  -f-  //)  is  of 
the  form 

(1  -f  A,"  +  A,.'^  +  etc.)  Aeie'J'  0* ,?/"  '^^  (^Vr  +fn'  +  m  +  IM). 

If  hi  the  last  factor  we  sul)stitut(:   the  preceding  ex])oneiitials  for  the  circular 
functions,  its  product  bv  ,V^',^V^'  in  the  case  of  a  cosine  reduces  to  half  of  the  sum 

(di)>  (ciir  (</>«)*  (W' +(;,/ (,';,/(x;  (^y. 

Substituting  the  values  of  tliese  expressions  in  t(>rnis  of  the  exponentials  just 
given,  developing  by  the  polynomial  tlieorem,  and  then  substituting  for  the  expo- 


GRNERAL    INTEGRALS    OF    T  L  A  N  E  T  A  II  Y    MOTION. 


3 


nentials  their  expressions  in  circular  functions,  wc  find  tlmt  this  sum  reduces  to 
a  scries  of  terms,  eacli  of  the  form 

COS 

''sin  ('•^^'  +  '^^-=+  •  ■  •  +'A.+i,?-'.+y2>.'2+  . . .  +./X), 
in  eacli  of  which  we  liavc 

'i  +  '^+  •  •  •  +  '■„=./+/ 
Ji+J.+  -  ■  ■  +./„  =  /■•  +  /.■'. 
The  expressions  A^e^  +  A^e'^  -\-  etc.,  comprising  products    and   powers  of  the 
sfpiarcs  of  r,  /■',  (ji  and  ^,'  by  constant  coefficients  by  the  substitutions  of  the  values 
(1)  reduce  tlieniselves  to  a  series  of  terms  of  tiie  form 

Ji  cos  (/,Xi  +  /A+  .  .  .  +  a„  +  .  .  .j\>.\  +J2A+  •  .  .  +i„>.'„), 
in  wliich 

/,  +  /,+  .  .  .+;,+^;+.  .  .  =0. 

By  tlicse  operations  and  by  corresponding  ones  in  the  case  of  sines  the  expres- 
sions to  be  integrated  iiually  reduce  themselves  to  tlie  form 


m'A 


SU) 


cos  ('"''  +  '■'  +  ''^-^  +  '""-^-^  +  •  •  •  +ii^'i  +  •  •  •  +i«?.',), 
in  each  of  which  the  sum  of  the  integral  coefficients  of  the  variable  angles  van- 
ishes, wliile  ^1'  is  a  function  of  the  mean  distances  and  of  tlie  '2n  quantities  A;  and 
Mi.  By  integration  this  expression  will  remain  of  the  same  form,  so  that  we  may 
regard  it  as  a  general  form  for  the  i)erturbutiun  due  to  the  mutual  action  of  two 
jJanets,  the  elements  of  each  being  corrected  for  secular  variations.  If  we  con- 
sider the  action  of  all  the  planets  in  succession,  we  shall  introduce  no  new  variable 
angles  except  their  mean  longitudes,  wliich  will  make  »  mean  longitudes  in  all. 
We  siiall  therefore  have,  at  the  utmost,  not  more  than  Sii  variabk;  angles. 

We  may  thus  conclude  inductively  that  by  the  ordinary  methods  of  apjjroxima- 
tion,  the  co-ordinates  of  eacli  of  -in  planets,  moving  around  the  sun  in  nearly  cir- 
cular orbits,  and  subjected  to  their  mutual  attractions,  may  be  expressed  by  an  infi- 
nite series  of  terms  eacli  of  tlie  form 


/.■ 


CCS 


•   ,i,;    ('l>-.    +    '2?-=  +     .     .     .    +   /3„?.3„) 


.  ;i2„  beinf 


»'„  ii  .  .  .  /;,„  being  integer  co(>ffici(Mits,  different  in  each  term ;  Xi,  ?vs  . 
each  of  the  form 

li-[-h,t 

/i,  /.J  .  .  .  /;,„  b(>ing  iiii  arbitrary  constants,  and  /;,,  h,  .  .  .  h^Jc,  being  functions  of  Su 
other  arbitrary  constants. 

We  sliiill  fnrtiier  assume  that  the  inclination  of  the  orbit  of  each  jdanet  to  tiie 
plane  of  .r y  is  so  small  that  the  co-ordinates  may  be  dev(>loped  in  a  convergent 
series,  arranged  according  to  the  powers  of  this  inclination.  wiiil(>  it  may  be  siiown 
tiiat  the  general  expressions  for  tiie  nvtangular  co-ordinates  will  be  of  tiie  form 


.1  ^  SI-  cos  ( /,>.,  +  /,>.,  -j-  ,  .  .  ^  ,;^,?..,j 
y  =  A7,'  sin  ( /,>.,  +  /,,X.,  4-  .  .  .  -f  /,„>.,„) 
z  =  Sv  sin  (y,>.i  +.y>j  +  .  .  .  +/,„>..„) 


(=i) 


G  K  N  K  UAL   I  N  T  E  G  U  A  L  S   OF    P  L  A  N  B  T  A  II  Y    AI  O  T  I  0  N  . 


Tlio  letter  <S'  being  used  to  express  the  sum  of  an  infinite  series  of  similar  terms ; 
/■•,  /,  and  y  having  the  signification  just  expressed,  and  each  system  of  values  of  the 
integers  i  and  J  being  subjected  to  the  condition 


h  +  h  +  h+  ■   •   •   •  +  ':, 


1 


(3)' 


j\+J-2+h+  •  •  •  ■  +iu  =  0 

It  is  evident  that  when  ■>•,  i/,  and  z  are  expressed  in  this  form,  any  entire  func- 
tion of  these  quantities  will  reduce  itself  to  the  same  form. 

W(>  shall  now  proceed  to  show  that  the  form  (8)  is  a  general  one :  that  is  to  say, 
that  having  an  approximate  solution  of  this  form,  if  we  make  further  approxima- 
tions, developed  in  powers  of  the  errors  of  this  first  solution,  every  approximation 
can  be  expressed  in  the  form  (3). 

We  can  make  no  general  determination  of  the  limits  within  which  these  approxi- 
mations will  be  convergent,  we  are  therefore  obliged  to  assume  their  convergency. 

§  '2.    Canonical  Trdusformatton  of  the  EQttationn  of  Motion. 

If  we  i)ut 

il,  the  potential  of  the  «  -|-  1  bodies,  that  is,  the  sum  of  the  products  of  every 
pair  of  musses  divided  by  their  nuitual  distance,  the  ditt'erential  equations  of 
motion  will  be  ;?(/t  -\-  1)  in  number,  each  of  the  form 

<1-X:       oil 
ill  vx. 

If  we  substitute  for  the  co-ordinates  themselves  their  products  by  the  square 
roots  of  their  masses,  putting 

.\i=W(-.r,;   Vj  =/»;//„  etc., 

tlie  ditt'erential  equations  will  assinne  the  canonical  form 

A\'e  suppose  the  index  /  to  assume  for  each  of  the  three  co-ordinates  all  values 
from  0  to  //,  tile  valiu;  0  referring  to  the  sun,  and  W(>  thus  have  3(//  -j-  1)  etinations 
of  tlie  form  (4)  the  integration  of  which  will  give  the  co-ordinates  in  terms  of  the 
time,  and  (!(//  -f"  ')  arbitrary  constants. 

\\  V  sliidl  now  (liniinish  the  numlier  of  variables  to  I)e  determined  in  the  follow- 
ing general  manner:  Su])pi)S(>  that  w(>  have  ii>  differential  ecpiations  of  the  first 
order,  l)etw(>en  ;//  variables  and  the  time  f,  each  being  of  the  form 


(It 


X 


Suppose  also  that  we  have  found  /•  integrals  of  these  equations,  each  of  the  form 

,/'(,»',, 3'.,  ....  ;'■„,./)=  constant. 

Let  us  assume  at  pleasure  )ii — /•  other  independent  functions  of  the  variables, 
each  of  tlie  form  ' 

£,  =  f/;, ■{..•,, a-,,, r,J\ 


GRXEUAL   i:\TEGllALS    OF    I>  L  A  X  E  T  A  1!  Y    MOTION  5 

so  thiit  the  m  variablo-s  x  can  be  oxpirs.siHl  as  a  function  ot  /.■  arbitrary  constants 
tiic  tinio  I,  and  t\ui  iii—k  variables  ' 

si!^2>  •  .  .  .  i;„i—k- 

Differentiating  the  above  expression  for  c.,  and  substituting  for  ^^'  its  value  X, 
we  sliall  have 

(U         vl  ex,  ^     -ex,  ^  ^     '"ox,; 
By  substituting  for  the  .-'s  in  the  right  liand  side  of  this  equation  their  expres- 
sions ni  terms  ot  ^1, £,„_.^.,  ^,  ,i„(l  the  arbitrarv  constants,  we  sliall  luive  the 

problem  reduced  to  tlie  integration  of  m~k  equations  between  that  number  of 
variables. 

In  tile  special  problem  now  under  consid(u-ation,  the  m  variables  are  the  co- 
ordinates X,  //,  ;,  and  tiieir  first  dcu'ivatives  witli  respect  to  tiic  time.  Tiie  integrals 
by  winch  we  shall  seek  to  reduce  the  number  of  the  variables  are  those  of  tiie  con- 
servation of  the  centre  of  gravity.  Wo  sliall  take  for  £„  c^,  etc.,  lin.-ar  functions 
ot  X,,  r,,  etc.,  so  chosen  tliat  tlie  reduced  ecpiations  shall  maintain  the  canonical 
form.     Let  us  take  tln'  //  -\-  1  linear  functions  of  tlie  co-ordinates  .<:— 

^0  =  «  +  fj(  =  a„oa-o  +  fXnXi  + -f  «;,^a;„ 

L  =«„oJo+ "„i'''i  + +«,„i''*'ni 

where  wo  have  put  for  symmetry 

w,  =  ca„„  or  Ho,  =      ,  (6) 

c  being  an  arbitrary  coefficient,  while  the  other  coefRrlents  urc  to  bn  chosen,  so 
that  tlie  resnltin;.  differential  equations  shall  be  of  the  canonical  form  Let  us 
represent  tiic  values  of  x  which  we  obtain  from  these  equations  by 

Differentiating  any  one  of  the  preceding  expressions  for  ^,  and  substituting  for 
jji  Its  value,  we  ii;jve 

<ii'     Wo '•■'•0  "^ "',  f 'J',  "^ '^w,ex„' 

If  we  suppose  ^,„  «•„  etc.,  n-placed   by  their  expressions   in  £„,  £,.  ot,,,  obtained 
liy  solving  the  equations  (5),  that  is,  by  tlieir  values  in  (7),  we  shall  have 

J  V  ^1  ^     -mil 

.  Substituting  these  values  iu  the  preceding  equation,  it  becomes 


I  ji 


O  E  N  li  11  A  L    1  N  T  K  G  11  A  L  S   O  V    V  L  A  N  E  T  A  R  Y    M  O  'I"  1  0  N. 

i/aioafo  _i_au«,i    I    WiL-Wia,  ."m",-,,  \  o'li 

"^V  w?o    "^    m,     '^'m,    ■+■ +    ,,^_   ;  ^^.^. 

In  Older  tliat  tliis  equiitiou  may  roducc  to  the  canonical  form 


f'aii 


^  oil 

dt-  ~  d£i ' 

it  is  necessary  and  sufficient  tlnit  the  expressions 

ojou^o  _j_  a>i«n    ,   «j2«,-2   , aj„a,„ 


VI., 


1)1, 


)llo 


111,, 


should  vanish  whenever  /  is  different  from  y,  and  should  reduce  to  unity  wlienevcr 
/=/.  In  otlun-  words,  it  is  necessary  and  sufficient  that  the  coefficients  a  sliould 
be  so  chosen  that  tiie  (m  +1/^  quantities 


]/  m^ 


(8) 


sliould  f(U-ni  an  ortliogonal  system.  The  first  line  of  coefficients  is  already  deter- 
mined by  the  equation  (G),  the  coefficiint  r  excepted,  which  is  to  be  determined 
bv  the  eoiulition 


+  :'+ 


7)1,,     '     ')» 


+;:=^' 


W'o  +  »'l  +....+  })!„  =  C", 


or,  from  (6) 

which  gives 

c  =  y  m, 

putting  m  for  the  sum  of  the  nuisses  of  the  entire  system  of  bodies.     Having  thus 


'Aw  = 

1)1, 

becomes 

y'o 

V''"i 

v/»'» 

\/ 1)1 ' 

/      1 

^z  1)1 

\/  ))i 

«10 

«ll 

«i» 

• 

l/»«« 

"■n\ 

««,. 

l/'"o' 

V<  ' 

*       '        • 

l/»»«' 

O  K  X  K  UAL    I  N  r  ]•:  U  11  A  L  S    O  P   P  [.  A  N  E  T  A  R  V    -M  O  T  1  O  X  . 


Tlio  lumihcr  of  coefficients  to  be  (letevniiiied  is  now  ii(ii  -\-  1).  The  total  niiiii- 
ber  of  conditions  nhicb  the  system  nmst  satisfy  is  ^'*-     -  /"J~-'\butoneoftliese 

beini,'  ahrady  satisfied  by  the  quantities  in  tiie  first  line,  there  remain  onlv  - '^"  "^  '^^ 

Z 
conditions  to  be  satistied  !)y  ii{n  +  1)  quantities,  we  have  therefore 

quantities  which  Jiiay  be  chosen  at  pleasure. 

The  f>("iieral  theory  of  tiie  substitution  Avhich  we  have  been  oonsidering,  and  tlu! 
various  modes  in  which  the  orthogonal  system  just  found  may  be-  formed,  have  been 
developed  very  fully  by  lladau  in  a  paper  in  AmiakH  ,h  V Kmlc  Xoniia/v  Supcniun; 
TomeV.  (18(58).'  We  shall,  ther(>foro,  at  i)resent  confine  ourselves  to  a  brief  indi- 
cation of  the  special  form  of  the  substituticm  whicli  has  been  found  useful  in 
Celestial  Mechanics.     AVe  first  remtirk  that  if  we  form  the  (/*  -j-  1)  equations 


by  giving  /  in  succession  all  values  from  0  to  n,  we  shall  have  by  the  theory  of 
orthogonal  substitutions  the  (/<  -|-  1)  equations 


If  we  sui)pose  in  the  first  e(piatioiis 
we  shall  have  from  (.5) 


whence,  l)y  substituting  tliese  valiK's  of  z,  and  //,  in  the  second  equation,  we  shall 
have  for  the  expression  of  ,r,  in  terms  of  f„,  £„  etc.  to  replace  equation  (7) 


\/  111 


-       1      "-21    - 

siH- 


IV: 


Tlie  first  t(>rm  of  this  expression  is  common  to  all  the  values  of  .,•„  represent iiiir, 
as  it  does,  the  co-ordinates  of  tlu!  c<'ntr(!  of  gravity  of  the  system.  It  may,  there- 
fore, be  omitted  entiivly,  wlieii  we  seek  only  llie  relative  co-onliiuites  of  the  vari- 
ous bodies,  and,  in  any  case,  it  will  disappear  from  the  differential  ecpiatious  of 
motion. 

The  most  simple  way  of  forming  tlie  coefficients  a,j  is  to  suppose"^""       of  them 

equal  to  zero.     L(>t  us  first  supimse  a,^  =  0  wlienevery>  /,  the  first  line,  in  which 
/  =  (I,  being,  of  course,  excepted. 

The  orthogonal  system  will  tlien  be  of  the  form 


'  Siw  lino  Tniii>i;iniuilioii  ik's  Kiiiiittimi.-  DiircivmifUus  ilu  In  DviiaiiiNniL'. 


8  GENKllAL    IXTEOllALS    OK    I'LAXKTAUY   iMOTlON. 


§/»'o        \/>lli        l/wio 


<^2()  «2I  «; 


l/Wo'      V"','     'i/ 


-^,    0,, 


0 
0 


(10) 


""»         "'II         "«2  a„„ 

Then  a„„  will  be  determined  by  the  condition 

?»„       ?/;  ' 

while  all  the  other  coefficients  in  the  bottom  line  will  be  determuied  by  the  condi- 
tion  ' 

/  4-  ~ —   -    =  0 

y  ?»,  «/„  '     w< 

Takin-  the  line  next  the  bottom  the  diagonal  coefficient  will  be  detennined  by  the 
equation  ^ 

a?,,  „_,  -\-  af,_,,  „_,      m„_i  _  ' 


while  the  remaining  coefficients  of  the  form  «„_,,,  will  be  given  hf  the  equ. 


itions 


The  gcMieral  values  of  the  coefficients  to  which  w.-  are  thus  led  may  be  expressed 
ui  tlie  loUuwnig  way :  put 

.".■  =  »'o  +  «'i  +  . 

by  which  m  will  become  ,«„.     Also,  suppose 


m 


m,. 


V,  = 


AVe  shall  then  have 


Vnj  .«^i  • 


^,2   _  »^  "/_! 

"i  , 

It  is  easy  to  prove  that  the  coefficients  thus  formed  f.dfil  the  required  eoncl.tions. 


m. 


f  l/vr. 


y  VI.,   / 


UKNIJUAL    l.NTKUUALS   OF   1' L  A  i\  KT  A  11  V    MOTION.  9 

Wo  sec  that,  suDposing  .i„  to  ix>i)resi,.iit  tlic  co-ordinates  of  the  sini  or  other  cen- 
tnil  bo.ly,f.  is  e.iuul  to  tlie  co-ordinate  of  the  Urst  phmet,  which  may  hi.  auv  one  at 
pkuisure,  rehitively  to  the  sun,  multiplied  by  u  func^tion  of  the  masses,  while  c  is 
efjual  to  tlic  co-ordinate  of  tlie  second  phmet  rehitively  to  the  centre  of  .'ravity'of 
the  sun  and  first  planet  multiplied  by  another  function  of  tlie  masses,  and  so  on 
1  lit-se  functions  t,,  when  divided  by  the  functions  of  the  masses  just  alluded  to,  will 
(lifter  ironi  the  co-ordinates  of  the  several  planets  relatively  to  the  sun  only  by 
quantities  of  the  order  of  magnitude  of  the  masses  of  the  planets  divided  by  that 
01  tiie  sun.  ' 

In  what  precedes  wc  have  considered  only  the  co-ordiiiat(-s  av     Of  course  the 
other  co-ordinates  are  to  be  subjected  to  the  same  transformation.     If  we  ivpresent 
by  ri  and  ^  the  corresponding  functions  of //  and  ^,  and  if  in  the  expressions  for  ^  r 
and  s  we  substitute  for  x,  v/,  ami  c,  the  expressions  (3),  those  quantities  will  tlu'in- 
selves  reduce  to  expressions  of  this  same  form. 


\  3.  Approximation  to  the  Rccpdred  Solutions  hy  the  Variations  of  the  Arbitrary 

Constants  in  a  First  Approximate  Solution. 
By  the  transformation  in  question  W(-  have  for  the  determination  of  the  relative 
motion  ot  the  n  +  1  bodies,  3«  differential  equations,  of  the  canonical  form 

vi-r  d('-  or:,'  Jt"--c>^,-  ^^^^ 

Let  us  now  suppose  that  we  have  found  approximate  solutions  of  these  equa- 
tions 111  the  form  (:3),  the  quantities  x,  //,  z  being  there  replaced  by  £,  ^,  ami  r. 
that  IS,  solutions  which  possess  the  property  that,  if,  on  the  one  hand,  eacli  expiv.- 
si.m  IS  twice  differentiated,  and  if,  on  the  other  hand,  the  values  (=})  are  substi- 
tuted m  the  second  members  of  (11),  the  two  expressions  shall  differ  only  by  terms 
multiplied  by  small  numerieal  coeffieients.  We  have  to  show  that  when'  w^  make 
a  further  approximation  to  quantities  o£  the  first  order  relative  to  these  coefiicients 
the  solution  will  still  admit  of  being  expressed  in  the  form  (3).  To  do  this  we 
sliall  make  the  further  approximation  by  the  method  of  the  variation  of  arbitrary 
constants,  remarking,  however,  that  the  usual  formula,  of  this  method  cannot  l.^^ 
appli.>d,  because  they  presuppose  that  the  first  approximation  is  a  ri.jorous  solution 
of  an  approximate  dynamical  problem,  while,  in  the  present  cas,>,  we  are  not  enti- 
tled to  assume  that  our  approximate  solution  (3)  possesses  this  quahtv  •  in  oth.r 
>yords,  we  are  not  entitled  to  assume  that  any  function  ri„of  the  quantities  s  ■,,  and 
„  can  be  formed,  such  that  we  shall  find  the  -.in  equations  of  the  form         "    ' 

rigorously  and    identically  satisfied  by   the    approximate  expressions,  both   with 
respect  to  the  time,  and  the  G.*  constants  which  the  solution  contains.     C'onse- 
qn.-ntly,  we  cannot  assume  tin,  (>xistence  of  a  p,u-turbative  function,  and  must 
emi)loy  other  expressious  iu  place  of  the  derivatives  of  that  function. 
We  set  out,  then,  with  the  three  sets  of  equations,  having  n  in  each  set 

2       November,  1874, 


10  OEiNEllAL   INTKGUALS   OF   PLANET  AH  Y    MOTIUX. 

I,  =  ^7.;,  cos  (/,;i,  +  i,?.,  + +  /,„a,„) 

)7,  =  *'/.•(  sin  (;,X, -f- ijXj  + +  '.m>.i,„)  (l^) 

^,  =  si/i  sin  (y,Xi  +p.a  + +j,,:k,), 

in  wliich  all  the  quantities  are  supposed  to  be  given  in  terms  of  G»  arbitrary  con- 
stants and  the  time,  each  ?.  being  of  the  form 

/(  being  an  arbitrary  constant,  wliich  each  b,  k,  and  /••'  is  given  as  a  function  of  3?i 
other  arbitrary  constants,  whieli  we  may  represent  in  the  most  general  way  by 

So  long  as  no  distinction  between  a  and  I  is  necessary,  we  may  represent  the 
entire  6m  arbitrary  constants  by 

Let  us  now  take  the  complete  second  derivatives  of  (12)  with  respect  to  the  time, 
supposing  all  (in  constants  variable.  We  shall  suppose  the  variable  constants  to 
fulfil  Lagrange's  conditions,  now  3«  in  number:^ 

^r,  'scij  dt~^'     jC,  ouj  at  ~  '     }t,  Siij  dt  ~ "'   ^  ' 

which  will  give 

dt  ~   dt  ~  ^  "  '"^*'- 

From  the  second  derivatives,  combined  with  the  differential  equations  (11),  wc 
shall  have  'Sn  equations  of  the  form 

i=i  vuj  at         O^i  iO-'' 

which  it  is  required  to  satisfy.     The  expression  in  the  right-hand  member  of  this 

ec^uation  corresponds  to  „ .  ni  the  usual  theory,  when  R  is  the  perturbative  function. 

Let  us  multiply  tiiis  equation  by  ^^^^ ,  and  add  up  the  3«  equations  which  we  may 

form  in  this  way  by  substituting  for  t^  all  the  values  of  -',  ■,;,  and  ;'  in  succession. 
We  may  thus  obtain 

*v  '  v""  '^^/  '':'*  da^  ^d^  _  '-?  o^  ^f, 
j=i    /^i  da^Oiij    dt        ddf,        i~i   r/2  iia,; 

the  sign  S'  indicating  that  all  values  of  y;  and  C  as  wc-ll  as  of  P.  are  to  be  included. 
The  right-hand  member  of  this  equation  corresponds  to  '.'  in  the  usual  theory. 
Let  us  now  multiply  the  equations  (13),  the  first  by  '}',  the  second  by  ^y',  and  the 
third  by  ^^^^' ,  and  add  together  the  3h  equations  which  may  be  thus  formed  by  giving 
I  ail  its  values.     If  we  subtract  their  sum  from  the  last  ecpiation,  putting 


UKiNKUAL   IXTKGRALS   OF    PLANETARY    MOTION.  H 

(==i  \ou,,  (jiij        dUj  da  J  ^ 

we  shall  have 

(".",)'';;'+(«.«.)'';;'  + etc_f  _  'v -^.^i,,  (i5) 

the  sign  ^  uicludiiig,  as  before,  not  only  all  values  of  /  from  1  to  >t,  but  the  cor- 
rcsijonding  terms  in  r,  and  'C,. 

]})•  giving  h  all  values  in  succession  from  1  to  6/t,  we  sliall  have  a  system  of  ()/* 
differential  eiinations,  the  integration  of  which  will  give  the  values  of  the  Km 
quantities 

iu  terms  of  the  time. 

IJy  tile  fundamental  assumption  with  wliich  we  set  out,  the  expressions  for  £, 
>;,  and  'C,  are  sueli  that  tiie  rigiit  hand  members  of  tiu'se  equations  are  small  quanti- 
ties of  which  we  neglect  tlie  powers  and  products.  We  may,  therefore,  after  solv- 
ing these  equations  so  as  to  get  the  derivatives  iu  the  form 

integrate  by  a  simple  quadrature,  sujjposing  r/„  <?,,  etc.,  iu  tlic  second  members  to 
be  constant.  jNIoreover  we  siiall  require  the  values  of  the  quantities  {a,.,  <i)  only 
to  the  first  degree  of  approximation,  and  witliin  this  limit  they  nuist  lu^cessarily 
conform  to  the  well-known  law  of  Lagrange  of  being  functions  of  tlie  constants 
oidy,  and  not  contahiiug  the  time  explicitly.  This  theorem  will  materially  assist 
us  in  tlu'ir  formation. 

§  -1.   Formation  of  the  Lagmnrjum  Coefficients  («„  a,),  and  Reduction  of  the 
E(/iuition/i  to  a  Canonical  Form. 

llestoring  th(>  two  classes  of  constants  represented  by  a  and  /,  we  shall  have  three 
classes  of  the  functions  souglit,  included  in  the  forms 

Let  us  now  differentiate  the  equations  (Li)  with  respect  to  the  time,  puttiu"  for 
brevity 

iA  +  '"A  4- +  !Jhn  =  h 

'.^.  +  '2^.  + +  i„:A,.,  =  N 

JA+JA  + +./;a„  =  6' 

y.>.i +,/2>...  + +;,„?,,„  =  iV'; 

we  sliall  then  have,  omitting  the  index  /  of  f>,  k,  luid  iV, 

£',  =  —  Shk  sin  .V 

>7',=       SJ>h  cos  A''  n,'-,') 

^',  =       Sh'k'  cos  N\ 

To  form  the  combination  («j.,  aj)  we  must  differentiate  the  equations  (1'2)  and  (15') 
witli  respect  to  r»,  and  a^,  and  substitute  the  results  in  (14).  Li  fbrininT  these 
quantities,  two  series  of  terms  represented  by  the  sign  S  of  summation  are  to  be 


I     1 


12 


U  K  N  K  II  A  L   I  N  'I'  E  a  II  A  T,  S   O  1'    P  L  A  N  K  T  A  U  Y    M  ()  T  ION. 


intiltipliod  tof^ctlu"!',  wliicli  rciulcrs  it  lu'ccssiiiy  to  be  iiion;  cxitlicit  in  vciJVcsciitiiifif 
tlio  (ioiiblc  Kiiinination  wc  tiius  cncouiitcr.  lliiviiifj;  ii.  of  ciich  of  tiic  (luimtitit's  x, 
y;,  iiiid  ^'  (listiiij,Miislic(l  Ity  writing'  tlu;  viirious  values  of  tlie  index  /,  wiiicli  takes  all 
iutej^er  values  from  1  to  ii,  tlie  quantities  h,  /.•,  and  X  siiould  all  bo  affeeted  with 
this  same  index.  Ihit  it  is  not  necessary  to  write  it  after  iV  or /<,  because  each 
N  is  common  to  all  the  £'s  and  >;'s,  or  to  all  the  i,"s,  respectively.  Again,  we  have 
as  many  values  of  .Vas  there  are  comltinations  of  the  coefficients  /„  /._,,  /,„  etc.,  which 
enter  into  it,  while  each  .V  has  its  corresponding  coefficients  /.',  /  in  number.  "NVo 
must,  therefore,  consider  /.■  to  be  written 

fu  {'.■>  '-i-,  ''a 'V.Ji 

while  h  and  N  arc  affected  with  the  same  indices,  the  first  excepted.  In  other 
words,  we  have 

/'  ('l,  i-l-.  '"a '■.„)  =  ''/'l   +  '7'3  + +  iJhn 

^Vl^  'V  'a 'ii..)  =  '"l''-l  +  ''i^-i  + +  '■M''-Su- 

Then,  in  the  sense  in  which  wi>  have  hitherto  used  the  sign  of  summation  <S'  we 
have  symbolically 

,=oc 


',  =oc 

i,  =  — OC     o  =  — OC 


V 


'3„  =  — OC 

To  avoid  the  comi)lieation  of  writing  so  nnuiy  indices  we  shall  represent  any  one 

combination,  as  (/,,  !.,, /;,„)  by  the  symbol  )■,  and  any  other  combination  by 

((,     Wu  shall  also  put 

I  =  n 

S'=  V    s. 
1  =  1 

This  summation  includes  all  the  terms  in  all  the  values  of  any  one  co-ordi- 
nute,  as  £,  y;,  or  s,  respectively.  A  sign  for  a  sumuuition  including  all  '.hi  co-ordi- 
nates is  not  here  necessary,  as  />•  and  A'  are  common  to  ^  and  y;,  while;  the  corre- 
sponding (pnuitities  for  if,  being  of  a  diffen-nt  form,  must  be  written  separately,  ^^'e 
have,  in  fact,  distinguished  them  by  an  accent. 

The  co-ordinates  and  their  derivatives  which  enter  into  the  expressions  (d^,  Uj) 
will  then  assume  the  following  form,  the  index  i  being  understood  after  k  and  A''. 

£i  =  '%/\u  eos  X^ 

ra  =  <V„/.'u  sin  X  (Hi) 

Ki  =  'V, /.•',.  sin  A"„ 

;",  ^       a:  (////),  cos  N', 


X.  —I.: 


t  sin  X,. 


V-  =  '^.'^  \  .     sni  AT,  -!-/.■    ,    •'  /  cos  X., 


da, 
da/ 


f  r)// 


sinA^;+Z'^^^'"/cos 


r)iu 


^s  A-,  I 


(17) 


OKXKRAL    INTIKJIIALS   OF    PLANETARY    MOTrON. 


13 


d 
da 


'  (IH) 


By  climiuiiijr  „,  into  „j  in  tlio  tlnrc  equations  (17),  nnd  makinf,'  tlic  reverse  elmii«e 
111  (IS),  HL'  liiivo  the  conipltte  expressions  necessary  to  form  any  term  of  tlie  v\- 
press  ion 

We  see  at  once  that  this  expression  will  be  of  the  form 

'i"   A%,,   I  -L,  sin  (y,  —  ,Y)  +  ,17  +  AT-  \ 

Since  the  expression  is  known  to  he  independent  of  /,  we  must  hav(>,  to  quanti- 
ties ot  the  first  (l(>frr,.,>  of  approximation,  A  =  0  and  A'  =  0  by  the  condition  that 
t,  r„  and  ';  satisfy  the  oriyinal  differential  equations,  and  the  coefficient  Au,v  must 
vanish,  unless  wo  have 

^u — iV;=  constant. 

The  co(>fficicnts  /*„  /,, ],,,„,  l)eing  supposed  incommensurable,  this  can  only 

happen  when  we  have  in  (:{)' 

'ly  =  'V  ;     V  =  'V,  etc., 
and  hcnco 

N^^N,,      - 

when  sin  {S\  —  X^  will  itself  vanish.      Hence,  (,,„  ,/.)  containinjr  no  constant 
term  whatever,  we  must  liave 

("*,  "j)  =  0.  (19) 

A!,Min,  differentiating  the  equations  (Ki),  the  first  three  with  respect  to  ?,  and 
the  last  three  with  respect  to  Jj,  we  find 

"'=       Sf^  {JJ.%  cos  N'f, 


=  _  >%  (i/>/,l  cos  N, 
=  —  S,  (JJblc),  sin  N, 
=  —  S,  iJjh'/i'),  sin  iV,. 


I    ! 


fl 


ii 


14  OKNKIIAL    INTKOHALH   OF    I'LANKTAllY   MOTION. 

From  tlicsc  expressions  it  may  be  shown  timt 

in  tlic  same  wiiy  that  \vv  tbund  {,i^,  «J  =(). 

We  liiive  next  to  consider  tiie  combiimtions  of  tiic  form  (u„  Ij),  for  which  tho 
expression  is 

i    I   I  (!u^  dij        i.lj  Oil,  ^  A<t  r/j  *^ 

Tin;  terms  wiiieii  do  not  eoiituin  t  as  u  factor  are  found  to  be 

-  -^'V.  {  OM),  ^^^  +  0A-),  J;l^^  ]  cos  i.y,  -  A-.) 
-i  'S;..v'.  I  oyn ^^;'^  +  Uj'^h  "^'^ ' }  ''OS  (x„  -  A",). 

<S"  havinjT  the  meanin;,'  ^iven  on  pa^'e  l'>. 

Tlie  only  non-periodic  terms  in  this  expression  will  be  those  in  which  /<  =  v,  mid 
these  terms  reduce  to 

or,  by  puttnig 

we  have 

(«,,/,)=_f^  ^2) 

These  (<xi)ressions  are  now  10  be  substituted  in  the  differential  ecpu.tions  repre- 
sented  by  (l.>),  wliich  will  then  divide  into  two  classes  aecordiuy  as  the  derivative- 

of  n  is  taken  with  respect  to  /„  /, or  /,„,  or  with  respect  to  «„  a, or 

<i3n-     Having  regard  to  equation  (20)  we  find  those  of  the  first  class  to  be  of  the 
form 

(/,  a,)        +  (/,, ,,,)      -  + j^^i.  a.  J  "'  -.  _  -f  _  v<  -f^.  -.i  _ 

If,  in  the  first  member,  we  substitute  for  the  coeflicients  their  values  cm,  noticiu" 
that  ^  " 

(J J,  a,)  =  _(„,,  /^.), 
and  in  the  second  member  put  for  brevity 

the  dia'erential  equation  reduces  to 

da,    ill  ^  da,  >/t  '^ ^"f9«,,„    (/<    -^^■'■' 

or 

'^  =  il,  (23) 


U  K  >'  !•;  It  A  I,    I  N  T  !•;  (1  U  A  I,  S    O  V   I'  I,  A  N  K  T  A  II  V    M  O  l'  1  O  N  . 


U 


Uy  ffiviiijf  y  all  viiliicH  in  Niuccssion  from  1  to  -hi,  we  sliiiU  liiivc  ;)n  ('(luiitious  to 

dctciiiiiiif  tho  viiriiitioiis  of  c,,  c.,, c,,,,  IVom  wliich  the  viiriiitioiis  of  «,,  «.,, 

....  .»<„,  lire  to  l)c  ohtiiiucd  by  tlu>  iin  ('(illations  ('^1).  Hut,  for  our  prcscut  pur- 
poses, it  will  be  more  convenient  to  consider  the  c's  as  the  fundamental  elements, 

and  to  consider  «„  n.,, « „,  to  be  replaced  by  c,,  r.,, c.,„  in  the  orijjinal 

ecpiations. 

The  second  class  of  differential  e(iuations  (15)  will,  by  (IJ)),  be  represented  by 

^        "  </>  (ft  ^  rd,      (-il  ol'   (:((,~  (.I- iki,~  <:i-(ii,\ 

Substitutiuf,'  for  the  coetHcients  in  the  first  member  their  values  (W),  we  sliall 
huvo  iiti  ecpiations  represented  by 

dih  ill  ^  i;i,,/t  ^ /.„,  "^  ,"i    \  ft'  in,  "^  "-"'•   } 

Putting  /•■  successively  equal  to  1,  "2 ;}</,  we  shall  have  3m  equations  of  tliis 

form.    Let  us  multiply  the  first  of  these  eciuations  by  ''-'',  the  second  by '.'"- ,  tin-  //// 


cc, 


(i\ 


(11, 


hy  .   ',  and  so  on  to  the  'iiith,  and  add  all  the  products,  uoticuig  that  the  theory  of 


functionul  deteruunuuts  ijives 


(  =3ll    "  ,     /■,, 


=  +  1  or  0 


according  as  /•  is  or  is  not  ctjual  toy.     Then,  by  jtutting 


we  shall  have 


dt 

ill, 

dt 

dl 


.111 
dl 


t 


(24) 


These  3»  equations,  combined  with  the  '^n  equations  (23),  will  give,  by  simi)le 
integration  by  quadratures,  the  p(n'turl)ation  of  the  (i/t  constants,  wliich,  being 
substituted  in  tlie  original  equations  (12),  will  gi\('  values  of  the  variables  which 
satisfy  the  original  differential  equations  to  terms  one  order  higher  than  they  were 
satisfi(Ml  by  (12)  originally. 

It  will  be  observed  that  if  our  functions  of  tho  time  and  (\n  arbitrary  constants, 
which  we  have  repres(>uted  by  f„  /■„  and  i!,'„  possessed  the  property  that  a  function 
lio  of  £>  'C,  and  :  could  be  found  such  tiiat  for  all  values  of  i 


t 
t  f 


10  (JKNKRAL    INTEGRALS   OF   I' L  A  >' i:  T  A  11  Y    MOTION, 

we  should  have  in  (2:3)  and  (24)  by  puttinju:  li  =  il  —  Liu, 

"^       c/j 

§  T).   Fiiiuhniuntal  liilafloii  hctu-ccn  the  Ciufficioitx  of  the  tiiiiv,  />„/>,.,,< Ye. ,  vouxhUicI 

ax  J'Hiirliaiix  "/' ('i,  <'j,  <tc. 

In  the  pveccdiiijj;  section  we  haxc  found  ourselves  able  to  express  the  first  approxi- 
mate vahas  of  the  variables  in  ti'rnis  of  ;J//  pairs  of  arbitrary  constants 


I  u 


'■'■Jn  'ail 

in  wliich  tlie  two  members  of  eaeh  ])air  are  coiijiii/ate  to  each  other;  or  possess  tiie 
l)roperty  tliat  tlie  expressions  (14)  all  vanish  except  when  o^  and  Uj  represent  tiie 
two  members  of  a  conjugate  pair,  in  wliich  case  we  have 

(/„  '■,)  =  +  1.  C^o) 

The  distinguisliinsi  characteristic  of  tlu>  intc^ifvals  we  liave  been  investigating  is  that 
tiiey  do  not  contain  the  time,  exc(>i»t  as  multiplied  by  the  '>]h  factors  /(,  wliicii  are 
functions  of  the  3/t  constants  o.  This  ciuiracteristic  will  enable  tis  to  deduce  a 
fundamental  relation  between  tiie  differential  coefficients  of  It  witli  respect  to  c.  In 
tlie  first  phice,  we  remark  that  each  c  has  a  />  to  which  it  stands  in  a  ))eculiar  rela- 
tion, ni  that  the  latter,  multiplied  by  the  time,  is  added  to  t!ie  /,  whicii  is  conjugate 
to  c  to  form  the  corresponding  '/..  Tiie  theorem  in  (pu-stion  is  this;  each  />  l)eing 
supposed  to  be  niar!<ed  witli  tlie  index  of  its  corresponding  c,  we  shall  have  for  all 
values  of  /  and  J  from  1  to  'Sn, 


in  other  words,  the  expression 

will  be  an  exact  difl'erential. 

It  is  ipiitt?  possilile  tiiat  this  theorem  may  admit  of  being  deduced  immediately 
from  tile  preceding  theory,  but  1  have  not  succeeded  in  doing  so,  and  have  there- 
fore been  obliged  t )  consider  tiie  jirolilem  in  the  reverse;  form,  ^^'e  have,  in  start- 
ing, sujiposcHl  ourselv(>s  to  liave  comjiletely  expressed  the  'Sii  co-ordinates  g,  y;,  i,',  as 
functions  of  the  Lvi  quantities 


«,.  II., 


and  we  have  just  shown  how  to  rejiiace   the  first  Wn  (pmntities   liy  the  quantities 
t\,c., fj„.      W  we  add  to  these  the  first  derivatives  of  the  co-ordinates  (1(5) 


G  E  N  E  UAL    1  X  T  E  (J  II  ALU   O  E    1'  1.  A  N  E  T  A  U  Y    M  O  T  I  O  N  .  n 

wc  shall  have  im  variables,  roprt'sentcd  by  c„  >;„  f,  ^\  rU,  Tf,  expressed  as  functions 
of  the  G/t  (|aantities 

Let  us  now  suppose  those  equations  solved  with  respect  to  these  last  (luantities. 
\Ve  shall  tlien  have  6/t  equations  of  the  form 

Ci  =  ^i\  K  =  'I'i.  whence  /,  =  i|',  —  ?*/,  (06) 

^  and  'P  being  functions  of  £,  r..  if,  etc.  Tlie  first  and  tliird  of  th(\se  rxi)r<'ssions 
arc  tlie  (i/*  first  intef;;rals  of  tlie  j;iven  equations,  or,  wliat  we  may  (ail  the  int(f,'nd 
functions,  bein<,'  tliose  functions  of  the  co-ordinates,  and  the  time,  which  rcuiai' 
etjual  to  arbitrary  constants  durnig  the  entiro  niovenu-nt. 

JiCt  us  now,  for  generality,  once  more  represent  the  ijn  arbitrary  constants  by 

"n  "•> '«,„„ 

and  let  us  consider  the  (()«)"  quantities  of  I'oisson  formed  from  the  "•eneral  ex- 
pression' 

the  symbol  ^\  including,  as  in  (14),  the  'in  values  of  £,  r,,  and  ^  in  succession.  Put- 
ting tiu'  giMieral  expression  (14)  in  tiie  form 


(..,.„,)=v;['.'-"-';----':=-"-]. 


forming  by  multiplication   tiie  jiroduct  of  tiiis  expression  l)y  (27),  then  puttinj. 
V  =y,  and  forming  the  sunuuation  • 


noticing  also  that  tlie  expression 


^  -, 


j  1  ( itjiii 

is  ecpial  to  unity  whenever  x-  and  y  represent  the  same  symbol,  and  to  zero  in  the 
opposite  case,  we  find 

an  expression  which  is  itself  ecpial  to  unity  wh(-n  u  =  !,  and  which  vanishes  in  all 
otlier  ( uses. 

Now  »»„  Oj,  and  »»„  may  here  be  any  of  the  (>/(  arbitrary  constants.  ],(-t  us  then 
suppose  ((„  a^  to  represent  /j  and  /^  respectively,  and  Uj  to  represent  Cj.  This  equa- 
tion will  then  become 

(',>  '•>)  [(«,  '■>]    \-  Oi,  '•.)  [/,.,  '•..]  f  (/„  '■,)V^,  r,]  4-  (<te.  =  1  or  0 


'   It  will  1)0  observed  llml  the  iiDtntioiis  iiitrddiu'cil  liy  I/npnuip'  iiiid  Poissou  rcspi'divet)-,  uro  hero 
reversed,  a  pripceeiliiiji;  wliicli  wiis  not  liileiilKiiiid  on  tlie  purl  of  llie  wiitor 
;t       November,  1874. 


>   (■ 


i 


f  r 


18 


GEX.ERAL   IXTEGRALS   OF   PLAXETARY  MOTIOxV. 


accoi-chng  as  l  and  ^i  repivsent  the  same  or  different  indices.     But  we  liave  already 
tound  that  tlie  .>xpressi„n  (/,  c,)  vanish(-s  whenever  /  is  different  from /,  and  reduces 

becom7  '     ""  '^'"''  "''^'''■'  "'"  ''^""^"     '^^''  "^"'''""'  '""  "'"  "'"^"•^•""S  thus 

[A'  f,]  =^  1,  (28) 

wliile  all  other  combinations  [/,,  cj,  [/,,  /.]  and  [c,,  <■,]  vanish. 

Let  us  now  return  to  the  integral  equations  (2(5),  and  first  form  the  combination 

'llw  conditions  (28)  therefore  give 

and  t"''"^^^^^ 

(29) 

the  first  equation  applying  whenevery  is  different  from  /,  the  second  when  they  arc 
the  same. 

Let  us  next  consider  the  combination  [f„  /.]  wliich  we  know  must  vanish  for  all 
vdue^ot  I  and  j.     forming  the  general  expression  (27)  from  the  integrals  (2G),  we 

[/,,  y  =  [<!;,,  ,,;.]  _  (    I  y,^^ ,,; .3 _  y,,^  ,,,_-,  |  _^  ^,  j-^^^^  ^^^^  _  ^, 

This  e,,„atioa  being  identically  zero,  the  coefficient  of  eacli  power  of  (  „„„t 
vanish  Identically.     This  gives,  in  the  case  of  tiie  middle  term, 

Forming  these  expressions  by  the  general  formula  (21),  and  putting 


we  find 


'•in     p-  ^   r,7 

i      L.  _I  (,f 

By  (2f))  all  the  terms  of  these  expressions  vanish  except  that  one  in  the  first 

Sle'fir?'    'h^'^'^  "f   '"'  "•"  •"  ^''•"  ^'^'^"'"^  i.-vhich/;  =  /,  iu  botho 
wluch  the  first  coeffacient  reduces  to  —l.     llcncc 


and  (;j())  now  gives 


^/^  __  fihj 


(31) 


GENERAL  INTEGRALS  OF  PLANETARY  MOTION. 


19 


§  «.  Development  of  D.,  D.J,  and  (^ J. 

Wc  have  next  to  find  the  forms  of  the  expressions  ilj  and  n',  which  enter  into 
the  equations  {'2'.i)  and  (24).     In  the  first  place  we  have 


n  =  v; 


Wi  111  J 


V{Xi  —  Xj)-  +  (/A  —  %)■-  +  (Z,  -Zjf 

We  now  substitute  for  a-,  y,  and  z  their  expressions  (9)  as  linear  functions  of  c, 
>7,  and  ^  respectively.  By  tliis  substitution  we  sliall  introduce  no  terms  of  tlie  form' 
0)7,  jtC,  or  'Ct.  Hence,  wlion  wc  substitute  for  £,  >r,  and  C,  their  expressions  in  infinite 
periodic  series,  tiie  reduced  expressions  will  contain  cosines  only.  In  fact,  usiu"- 
the  forms  ° 

^i  =  SK'i  cos  iV 
r,i  =  Sl'f  sin  ^V 
<:,=:  A7/,sin.V', 
we  shall  have  from  (12)  when  we  put  for  brevity 


C"-"'0^'+("'~"-^>'  +  «tc. 

XWij  IHj  I  \?H,.  9»^.  /    -     ' 

•»•,•  —  Xj  :=  .S7",j  cos  iV ; 
2/,  — .'/,  =--  'S'Z-,^  sin  iV; 


=  * 


(jf 


(32) 


\t>\2 


2,  —  2,'-  z=  A'/.',.,  sin  iV. 

Each  d<>nominator  in  H  will  therefore  assume  tlie  form 

^/  ( A7-  cos  .V)-  +  (,S'A-  sin  A')-  -f  (A'A-'  sin  .V' 

Wlien  W(<  form  these  three  scjuares  we  find  that  every  term  of  th(-  form  //  cos 
(-^u  +  ^^)  1"  till'  fii'st  s(iuare  is  destroyed  by  a  corresponding  term  — //  cos  ( .V«  -I-  K.) 
in  tii(>  second  square.  Hence  the  sum  of  th(-se  two  squares  will  only  contain  ternis 
of  the  form 

;,  cos  {N^  —  X,). 

Since  in  each  value  (IT))  of  .V  we  have 

'\  +  !:  +  h  + -\-L  =  h 

we  sludl  have  in  X^  —  iV, 

Also,  since  in  .V  the  sum  of  these  coefficients  is  zero,  it  follows  that  the  smn(< 
thins,'  will  liol.l  tru(>  of  th(>  third  of  the  precedinj,'  s(iuares.  The  denominator  in 
question  may  therefore  be  expressed  in  the  form 

in  wiiich  each  A'^  is  of  the  form 


where 


«.  +  '■■  +  (,  4- +  'a„  =  0. 


mmmmt^^- 


J±:J..M:^.^d^y,j0^ 


f  y 


f  r 


20  GENERAL  INTEGRALS   OP   PLANETARY   MOTION. 

The  possibility  of  developing  the  reciprocal  of  this  denominator  in  the  usual 
way  depends  upon  the  condition  tluit  the  constant  term  of  *S1-  cos  N  is  larger  than 
the  sum  of  the  coefficients  of  all  the  other  terms,  a  condition  which,  so  far  as  we 
yet  know,  is  fulfilled  by  all  the  planets  and  satellites  of  our  system.  Representing 
tliis  constant  term  by  1,-^  and  the  quotient  of  the  dum  of  all  the  other  terms 
divided  by  k^  by  A,  so  that 

SI;  cos  iV  =  ^•„(  1  +  A) 
the  developed  expression  for  fl  will  be 

n=s'';;^(i-iA+i-|A»-etc.). 

When  we  develop  the  powers  of  A  this  equation  will  reduce  itself  to  the  form 

n  =  Sh  cos  (/,;i,  +  L?.,  + 1,?.3  + -(-  i3,.x,„),  (33) 

each  X  being,  as  before,  of  the  form 

,.,    .  ^  ^i  =  h  +  ht, 

while  in  each  term 

h  +  h  +  h-\- +  hn  =  0. 

To  form  the  seco  id  part  of  ilj  and  of  il)  in  (23)  and  (24)  we  have  to  differen- 
tiate the  expressions  (12)  twice  with  respect  to  the  time,  and  once  with  respect  to 
the  arbitrary  constants  wliich  enter  into  them.     Putting,  as  before,  for  brevity, 

^=*l^l  +  *'2^3+ +hn^3n 

b  =  iA  -\-iA-\- +  HnK, 

we  have 

^•2  =  —  Sh^Jci  cos  N 

^<  = -SIH;  sin  N  (34) 

^^,*=  —  Sh"k',smN'. 
or 

For  the  other  derivatives  which  enter  into  il'j  we  have 

^1  ^  —  SI:  k,  sinN 

^f  =      Sijki  cos  N  (34)' 

.>,'=      SJjl:'iCos  N'. 

Forming  tlie  sum  of  the  products  which  enter  into  11^,  in  the  manner  represented 
in  §  4,  it  becomes 

^r  S^S,  ^(  !jl-:) , {h%,^  sin  (N.  —  JV;) 

+  UjA'i\{f>''l'''<)^{^^n(N',  —  N'^)-sin(N',-\.N'^))\.    (35) 


OENKllAL    IXTEGKALS   OF   PLANETARY   MOTION.  21 

This  expression  reduces  to  the  form  S II  cos  N,  where  in  each  value  of  iVwe 
have 

li  =  0. 

In  tliis  expression  it  may  be  worth  while  to  give  the  complete  value  of  ^corre- 
sponding to  any  value  of  N.  The  value  of  the  latter  is  comphstely  determined  by 
the  indices  /„  i,,  etc.,  which  multiply  X,,  ?.„  etc.,  in  its  expression.     Let  then 

represent  the  value  of  N  for  whicli  we  wish  to  find  the  corresponding  value  of 

mii'J.i /,„)  by  means  of  (35).     The  required  term  will  be  found  by  taking 

in  (35)  all  combinations  of  )•  and  /i  for  which  we  have 

N,-N^=  N, 

N\.  —  .v;  =  N, 
or  .v;  -f  iV;  =  K 

I-ct  us  represent  the  combination  of  indices  v  in  iV"    by  k„  L,  vtc,  and  those  in 
N\   byy,,y^,  etc.,  so  tliat  we  have 

iV,  =i,?., +y,?.,  + +y:,„X3„. 

Then,  in  order  that  tlie  sum  or  difference  of  these  angles  and  of  iV^  may  make  .V, 
according  to  tlie  formula;  just  written,  we  nmst  have 

-X'^  =  (,"1  -  'l)'<.  +  (,«2  —  «.)>.2  + +  («,,„  —  i3„)?.3«, 

and 
or 

^V  =  ('l   -  i,)>-.  4-  ('2  —  .?;)>...  + +  ihn  —J,J>.,n- 

For  the  corresponding  coefficients  of  the  time  h,  we  have 

^v  =  (,".  —  h)^  +  (ih  -  i^h,  + +  («,„  _  ,;„)7>,„ 

?-;  ±  (.A — A)^.  ±  a  -  Q\  ± ±  (i„. — i.u,)K. 

Affecting  k  and  I'  with  the  proper  indices,  as  explained  in  §  4,  tlie  part  of  the 
coefficient  IIj{!,,  /., /,„)  corresponding  to  any  one  value  of  tlie  angle  N,,  will  be 

i  ^  n 

^Xjljki{lii,^l„_, )  A-j(,M,— «i,/<2— <o, )7>^- 

+  J f^jj^^'U^  ./='  Wm  I ^■',(./.-'.,.y;.-'i, )—/.•',('■,—;„  'o— i, ) I 

where  the  values  ofh^  anu  /^  are  those  just  given.    The  complete  value  of //;(/„  !.,, ) 

will  be  found  by  taking  the  sum  of  all  the  terms  which  we  can  form  by  giving  to 

;/i,|(*o,  ctc.,ji,j.,, y„„  in  these  expressions,  all  admissible  combinations  of  values, 

that  is,  the  complete  expression  will  be  given  by  writing  before  the  first  line  the 
symbols 

|Ui=OC  A«2=0C  |Uj„=OC 

2  2        2 

i"i  =  — OC         !«,  =— oc  |U3„  =  — OC 


j' 


22 


GENERAL   INTEGRALS   OF   PLANETARY   MOTION. 


V 


CM) 


and  before  the  second  one 

y,=oc  y,=oc 

.2  2        ...  _ 

i,  =  -cx:        ,72=— oc  .;3,>  =  -oc 

Differentiating  {US)  with  respect  to  /,,  wo  have 

By  the  substitution  of  these  expressions  (23)  now  assumes  tlie  form 

(ft V/jSm.V,  (;57) 

putting  for  brevity 

h'  =  ij7i  -)-  If., 

By  the  fundamental  Iiypothesis  tliat  tlae  adopted  expressions  for  £,  .,  and  C  are 
(.  -.)  and  (3(,)  all  the  terms  wh.eh  are  not  of  the  order  of  those  neglected  in  the 

n  le^eHlT  r"  '  """''  ''^"''  ^°  '^'''  ^'  ''  ''  '''"^  -•^"-  «^'  ^'-'  ^--'ities 

neglected  m  that  approximation. 

To  form  the  equations  (24)  we  differentiate  (12)  with  respect  to  .,  wlu-rel)y  omit- 
Un,^  mdex  .  wUh  which  .,  „  ^,  Z,  and  ,  are  always  to  be  considered  as  :^:d, 

—  A        cos  A  +  /  ,V^  .    sin  A^ 


dc 


f'C: 


cc 


'i  „  OK    .      ,,  /:f) 

,  =  aS"  .     sm  X-{-tS/,-  .    cosiV 

?:  r,f,  '  : ,, 


vc 


(iny 


The  sum  of  the  products  of  these  expressions  by  (:3-t)  which  enter  into  (24)  is 
-\  l'[  S-^,„  I  (i=/.)^  J^  ,0,  (^;  -  ^)  -  ^  ilrA%  ^^'>'  «in  ( a;  _  A^) 

+  l(i''A-%  ^J-  (cos  ( at;  -  A^;)  _  cos  ( A^;  +  A^;) ) 
i  /  (^'^A04^' (sin  (A;- a;)  _  sin  (A-;  +  A^;)  I , 


while  by  differentiating  (;33)  we  find 


.;^=A(.^cosA^_/^.^^si„x).  (07)" 


Taking  the  difference  of  these  two  expressions,  the  equations  (24)  will  assume  the 


\ji  =—Sh"  cos  .V+ 1  Sir  sin  .V. 


(38) 


the  qmmtities  J,  and  r  being  formed  by  a  process  similar  to  that  used  in  forming 
n.     We  have  now  to  mtegrate  the  expressions  (37)  and  (3S),  and  substitute  th^ 


O  E  N  K  11  \  L    INT  K  G  K  A  L  S   OP    P  L  A  N  E  'I'  A  U  Y   AI  OTIO  X .  23 

resulting  values  of  <■,  and  /^  in  the  expressions  (12).  Representing  the  perturha- 
tions  of  eacli  quantity  by  the  sign  h,  we  shall  have  to  increase  each  value  of  X  by 
the  quantity 

^?.,  =  ,M,  +  thbi. 

We  here  have  the  time  t  outside  the  signs  s!n  or  cos  in  both  (V,,  from  the  integra- 
tion of  (;38),  and  in  (hlji.  We  must  next  find  the  sum  of  tlie  terms  thus  introduced 
into  b?.i.     Dilfereutiuting  this  expression  we  have 

We  have  now  to  form  the  sum  of  the  terms  in  the  seeond  member  of  this  ecpiation 
wiiich  are  multiplied  by  /.  iJeginniug  with  the  second,  we  have,  omitting  the  in- 
dex of  I) 

dh ch  dc,    .   c/>  dc. 

dt  ~dc,dt  '^  dc.dt  '^  ^^^' 

Substituting  for      '  their  values  in  (31),  this  equation  becomes 

dh  (      dh  ch    .  ,    ^     ih    \    .     .. 

which,  after  multii)lying  by  t,  is  to  be  added  to  the  last  member  of  (38).     But  it 
will  be  more  convenient,  instead  of  using  h  and  It"  in  these  expressions,  to  retain 
T^C      /-  7-''* 

the  expressions  ,!?,',,',  and  ', !?  in  their  present  analytical  form,    lleprcsentina'  them, 
at    dt-  dl'  ^ 

for  brevity,  by  t",  y;",  and  ^",  the  equations  (23)  and  (24)  become 

dcj  _      eq      .■  = »  i  ^„  cJi,        „  fV*  ,    .„  o^s  ,•  1 

dt  ~    eij     ,r,  t  ^  *^  "^ '^  'eij  ^^'efjl  (4«) 

dt        dcj  ^  .• , .  1  ^ '  dcj  ^  '^  '■  c-;^  ^  ^  '•  6cj  i " 

If  in  the  first  of  these  equations  we  substitute  for  the  derivatives  their  values  in 
(34)'  and  (3(>i,  it  becomes 

'J  =  -  ^'  {  '^''  -  -  (^'"'^''■')  }  ^'"  ^^  +  -  (^"^'J  ^'^  ^"'^  "^'  +  -  (^'".A  /'•'.)  cos  ^"• 

Substituting  in  the  first  of  the  above  expressions  for      ,  we  have 

dt 

'^''  —         V  /  .  Sh  oh  '  ^     .     rlh        )  ,     .      „ 

^    dt—'n      'v.,+'^,.c,+ +'-,.e:  [^^'"^ 


j 


i  1 


34  GENERAL   INTEGRALS   OF  P  L  A  N  t  i  .\  R  Y   MOTION. 

We  have  next,  in  tl>e  second  of  equations  (40)  to  substitute  tlie  expressions  for 
the  derivatives  in  (37)'  nud  (37)",  nitaininy  only  the  terms  multiplied  by  I.  This 
gives  by  substituting  for  b  its  developed  expression 

b  —  ij)i  -\-LJ)i+ +  '■;.,>  K 

+  ,\^,{:,^  +  J^+ +  ,.«.■)}    e„»^ 

+  .{.r^',(^^^  +  ^*+ +i,*.)}    .^K. 

Adding  this  expression  to  (41),  we  find  that  the  sum  reduces  to  a  series  ot  .ms 
eacli  of  which  has  a  factor  of  the  form 

6b,_dhj 

By  (31)  these  factors  arc  all  zero.  Hence  the  terms  of  (39)  multiplied  by  <  destroy 
each  other,  and  we  have 

the  parenthesis  around  ^  '  indicating  that  all  the  terms  multiplied  by  the  time  in 

that  expression  are  to  be  omitted ;  in  other  words,  that,  in  taking  tin;  derivatives  of 
n,  ^,  y;,  and  ^  with  respect  to  o„  we  are  only  to  consider  the  coeffici*  iits  h,  k,  and 
k'  as  functions  of  these  quantities,  and  are  not  to  vary  i,,  b,,  etc. 

§  7.  Form  of  (he  Second  Approximation. 

The  rest  of  our  process  is  now  as  follows :  By  iutegratii.j,'  (37)  and  (38),  the 
last  member  of  (38)  being  omitted,  we  have 

Scj  =  S^'-'cosiV 

(,y.)  =  _  S^''j  sin  N. 

The  co-ordinates  £,  y;,  and  s  in  (12)  being  expressed  as  functions  of  the  quanti- 
ties Cj  and  /j,  we  are  to  suppose  these  quantities  increased  by  their  perturbations, 
that  is,  we  are  to  lind 

^^  =  24^^  +  24^^' 
or,  since  we  have  replaced  /,  by  ?.<, 


0  K  N  E  UAL    1  N  T  K  (J  U  A  L  S    O  V    V  L  A  X  10  T  A  U  Y    M  U  T  J  O  N . 


25 


111  (4;J)  wc  liuve 


and,  integrating, 


J  =  :\n 


hb,=^'p  ,'.cj=,S   X   7': 


/(';  (b. 


oc, 


COS 


iV, 


>-i    0  ov. 


^^^<  =  ('V,)+J'Vv/^ 


f/' 


j    -  3/1 


/(',  (7/ 


\   6 


;-   I    0-  cr, 


I 


iiu  iV, 


which,  for  brevity,  wo  may  represont  by 


h'/.i  =  «S',7v,  sin  iV. 


(44) 


l)iitting 

Tn  adding  tho  ofroct  of  tho  ix-rtuvbiUioiis  hci  to  £,  >;,  and  ^,  wl-  iiir  to  vary  only 
/•,  tile  fxpressions  for  h^,  etc.,  being 

<V  ='S.  I  hk  cos  .V—  /•  sin  N{IMx  +  ip...  + +  /;„A>.;,„)  [ 

h  =  '%  I  '^^-  .sin  N+  k  cos  .V(/,,^>.,  -f  Uv,.,  + +  4„,S>.,„)  [ 

W(  =  S,  I  a-'sinXf //cosXcy,,^?..  +^V^X2+ +iJ>.J  I 

We  are  to  put  in  these  expressions 

and  the  values  of  (^?.  in  (44).     We  tiius  find 

+  ^  «^<.  {  i:.  ( ^'  ^J| ),  -  Z^.  (/,/.,  +  /,A,  + +  /,„A,„x  I  cos  ( a;  _  ^;) 

;i,  =  1  s\,,  I V,  (^'  ;:J' ),  +  k^  {i,L,  + ;,/.,  +  .;...+  /,„z,„),  I  sin  (X  +  X.) 
+  ^  ^n,..  I  :^.  (^^  ;;'■ ).  -  A-,  {lA  -V  u.  + +  /:,„A„),.  I  sin  ( a; - X,) 

5C  =  i  '^%|:t.(^'^|),  +  ^<„(.^Z,+^A,+ +./.Z.J.|sin  (X',  +  N.) 

+  h  ^^.  [  S<  (^'j;  ^),-l/^J,^  +.^A,  + +i<„Z,„X  I  sin  (A-;,  -  N,) 

Since,  in  iV„  we  have  'ii=  1, 
while  in  ^V,    "      "      ii  =  0, 

4         November.  1874. 


'f 


it 


26  GENKllAL    IXTEGllALS   OF    PLANETAUY    MOTION'. 

it  follows  tlmt  all  these  terms  will  be  of  the  same  form  with  those  already  contained 
in  t,  y;,  and  s  (12). 

In  the  preceding  inte<,'ration  we  have  tacitly  siipposed  the  coefficient  of  the  time, 
b,  never  to  vanish  in  any  case,  lint  some  of  the  values  of  A'  will  necessarily  he 
zero,  and  in  this  case,  instead  of  having 

JA(//cos jy'=  .'  sin  iV, 

we  must  put 

fi  (It  cos  N  =  Id. 

The  only  terms  of  this  form  are  found  in  hi.  If,  in  (3S),  we  represent  the  coeffi- 
cient of  the  vanishing  term  by  /("„,  we  shall  have  for  the  terms  in  question 

-V  =  -  rj. 

This  adds  to  '/.  the  same  expression,  and  is  equivalent  to  diminishing  h  by  the 
quantity  /("„.  We  make  this  chang(>  not  only  in  the  original  terms  of  £,  r,,  and  'C, 
but  also  in  tiie  terms  of  (S^,  (V,  and  (^^',  because  the  change  will  only  affect  them 
by  quantities  of  tlie  second  order,  which  we  have  rejected  throughout. 

Making  these  changes,  the  expressions 

i  +  Ki       r  -\-  'Vi       "i»l  if  +  K, 

will  now  satisfy  the  differential  equations  (11)  to  quaniities  of  the  second  order, 
while  their  Ibrni  will  still  be  in  all  respects  the  sani(!  as  in  (12).  As  we  have 
made  this  one  approximation  without  changing  tlie  form  of  tlie  original  integrals, 
so  may  we  make  any  number  of  successive  approximations.  We  may,  therefore, 
regard  the  form 

e  =  ,S7.-  cos  (;,?.,  +  ?j?.2  4- +  /;,„?.:,„) 

y;  =  aS7.-  sin  (;,?.i  +  *,?.,  + +  l,„X,„) 

(,'  3=  S/i  sin  (./,Xi  -\-jnX.,  + +j,n>-jJ, 

where  each  '/>.  is  of  the  form 

?..•  =  '.-{-  bA 
7,  being  an  arbitrary  constant,  and  /.',  Jc,  and  b^  being  each  functions  of  3)t  other 
arbitrary  constants,  while 

'■|  +  h+ +  '■),.  =  1, 

fin<l  ./i  +.h  + +.hn  =  0, 

in  each  separate  term  under  the  sign  S,  to  be  a  genc-ral  form  in  which  the  relative 
co-ordinates  of  n  planets,  revolving  in  nearly  circular  orbits  with  a  nearly  uniform 
motion,  may  be  developed  when  the  approximations  are  continued  indefinitely. 
This  may,  therefore,  be  regarded  as  the  general  form  of  the  integrals  of  planetary 
motion. 

§  8.    Geuernl  Theorem, 

If  irr  I'xprpxx  the  rcldti'rp  lirinp  force  of  the  rnt'nr  xt/strDi  I'li.  Icriiift  of  the  ediionlrol 
ehnunlx,  tlir  eoejjh'irutu  i,f  the  time  b„  b.^, /<„,  irlH  each   be  eqiinl  to  the  negative 


G  K  N  K  UAL   I  N  T  E  0  U  A  L  S   O  V   PLAN  E  T  A  II  Y   M  O  T  1  O  N . 


27 


of  the  ilrr! rat! I'll  d/  the  coHKiunt  tvnii  of  the  11  riiKf  force  with  renpert  to  Itx  corren/iojiiJ- 
iinj  canonical  element.  Tluit  is  to  say,  it'  we  rcpiL'scnt  the  foiistiiiit  term  oi"  the  iiviiiy 
force  by  V,  and  supiiose  V  to  be  expressed  in  terms  of  the  ciinouicul  elements,  we 
bIiuU  huvo 


h=-: 


■■V 


dc. 


From  the  expressions  (9)  for  a-,  and  tlie  rorresponding  expressions  for  i/  and  z,  it 
will  be  seen  that  the  expression  for  the  relative  living  force  is 


^\./v,.^'  +  ./nJ'^ ) 


V"'u"'    '    V 


+  ^ivk^^+v<'-^ y 


t 


-\-        etc.  etc.  etc. 

-)-  corresponding  terms  in  r'  and  l,'. 

Here  the  coefficients  of  t',  etc.,  are  those  which  we  have  sliown  to  form  an  ortho- 
gonal system,  and,  by  the  properties  of  sncii  a  system,  the  expression  reduces  to 

^i;.(i'^  +  -^''.  +  s'^)- 

.Substitutuig  for  ^',  r^,  and  ^"  their  periodic  expressions 

-  Sbk  sin  N 
Shi-  cos  N 

the  constant  term  of  the  living  force  is  fonnd  to  be 

the;  sign  S*  having  the  signification  given  on  pnge  12.  Compare  tliis  expression 
with  tiiat  of  e,  in  {'21).  Multiply  each  f,  by  its  corresponding  b^,  and  add  all  the 
vroducts,  remembering  that 

I)  =-  !J|^  -If-  !.,!).,  -|-  etc.  for  ^  and  /;,  aod 
()  =JJt^  -\-,)A  +  <'t''.  for  V 
We  thus  find,  from  the  expression  for  l''just  given, 

2  V=  h^c^  +  h.f.,  +  /|,c,|  + +  6;,„e,„. 

Differentiating  this  expression  with  respect  to  c^  and  substituting    '  '  for    - 

ccj         re; 

W'e  have 


5fo 


+  '•;!« 


f)6, 


(46) 


We  have  now  to  show  that  h  is  a  homogeneous  tunction  of  the  degree  — 3  in 
(c„  f',, c,„).     Let  us  represent  such  a  function  of  the  nth  degree  by  ['•"  ] 


:1 


'It 


* 


Aiirify 


98 


0  (•:  N  K  I!  A  F,    I  N  T  i:  (i  I!  A  I,  S   ()  F    1. 1,  a  N  K  T  A  i:  V    M  o  T I  O  N. 


Let  us  r.>i)r<-sn,t  the  li „•  ..l,.n,..Mts  ..f  tl..  systr...  l.y  r^. .,,,  vtc.     Sinro  u-,  y,  2,  a.ul 

«,  >:,  s,  ill-'"  Mil  liiuiir  .o-..i(liiiutc.s,  »-,.  Imvr  in  ili,.  (.xpirssioiis  (1(1)  of  tlit"  latter 

Kv.iy   tin,.-   Nv..  .lifH.rcutiiite  th.-se  ••xpicssions   with    resp.Tt   to   the  time    we 
miiltiply  the  eoifficieuts  by  h,  u  linear  liiucticm  of  />„  I,,,  cte.     Ik'uce 


:  =  [""\6'^'j. 


'I'lic  form  of  the  potential  il  shows  tlmt 

:/;i;:'\:':::;':.."'';r,™  ""■  '""■  '"■  '""■""'"■■ "'°'-"""»'  *° "-  ■"-■™  -i- 

In  order  that  the  diftWeutial  e,,uation  ;^f  =  ^^  n..y  he  satisfied  identically  we 
must  have 


or 


K",  6'-"]  =  [„'--■)], 


The  expression  (-^1)  for  ,•,  /.•  hrinfr  Unrar  in  a,  is  of  the  form 
Ileue..,  wh.M,  wt>  express  A,  in  terms  „f  r„  <•„  cte.,  we  must  have 
Tiie  fundamental  property  of  homo,    neons  functions  now  gives 

Substituting  in  (4G),  we  find 

t/c, 
which  is  the  tlieorem  enunciated.  '* 

This  theoren.  camiot  b,.  .lirectly  employe,.   ,„  ohtuin  the  values  of  /,,  for  the 
eason  tl.at   1    cannot  be  .Ictern.ined  as  a  InnCou  c.f  the  canonical  constants  until 
the  equations  ot  motion  are  completely  iiitei.'r  .te.i. 

§9.   Snmmnri/ of  UrmUK 

The  following  is  a  brief  summary  of  some  of  the  results  which  follow  from  the 
preceding  nivestigation. 

We  first  suppose  that  we  have  found  expressions  for  ,",  ,,  and  ^  of  the  forn,  (!■>), 
.uch  as  ulent.cally  satisfy  the  .lifferential  equations  (11).     We  also  conceive  the 


i 


(J  K  X  !•;  u  A  li  1  N  r  !•;  o  ii  a  i,  s  o  i    i- 1,  a  >  k  r  a  ii  y  m  o  t  i  o  n . 


29 


(luiuifitifs  /.■  and  /*  as  cxpn  ssed  in  terms  of  '-in  t'anonical  coustauts  t„  f^,  f, 

t.,„,  so  chosiin  tliat  tlio  fxpriNsion 

I     I    I  (Cj  f/j.        fc^  i/j         ffj  ell,  J 

Ninill  rcilncc  to  unity  wlicii  k—J,  tinil  sliall  vanish  whenever  any  ot'er  of  the  Cii 
(iniintities  '•, r„„  /, /,„  is  substituted  tor  /j.     'I'iien:  — 

Tlici'.itn    I,  —  If,   taking  the  entire   si'ries  of   ',iit   co-ordinates  represented    by 

£, i',,,  r^i >:.4.  s'l Cfu  ^ve  multiply  the  square  of  each  toetticient 

k  l)y  the  coefficient  of  the  time  in  the  correspond injj  angle  /jXi  -j-  I'a?.,  +  etc.  (that 
is,  by  the  corresponding  (piantity  !',<»,  -\-  ih^  -j-  etc.,  ory,/*,  -\-JJ>i  -\-  etc.),  and  by  the 
coefficient  !j  or  j)  of  any  one  of  the  ?.'s,  as  >.j,  which  '/.  is  to  lie  the  same  tiiroughout, 
then  all  tiie  constants  c,  except  r^,  will  identically  disappear  from  the  sum  of  all 
these  products,  which  sum  will  reduce  identically  to  '-.V^.  I'his  theorem  is  expressed 
in  eipiatiou  ('21). 

T/i'DiTiii  II. — The  'An  coefficients  of  the  time,  6,,  tj,  etc.,  considered  as  functions 

of  ci,  t'a,  etc.,  fulfil  the  '         ,,  —   conditions  expressed  by 


vh, 


/'■''> 


where  /  and ./  may  have  any  values  at  pleasure  from  1  to  3n.  They  arc  therefore 
all  till'  partial  derivatives  of  some  one  function  of  c,,  v., c,„. 

'J7wuiT)ti  111. — This  function  is  the  negative  of  the  constant  term  of  the  expres- 
sion for  the  living  force  in  terms  of  c,,  c,.,,  etc.,  as  shown  in  the  last  section. 

'f/iioniii  IV. — The  sum  of  the  canonical  elements  c,,  Cj c,,,  is  equal  to  the 

"  constant  of  areas."  this  constant  being  cither  the  sum  of  the  canonical  areolar 
velocities  on  the  plane  of  A""!',  or,  which  is  the  same,  the  sum  of  the  products  ob- 
tained by  multiplying  the  actual  areolar  velo(  ity  of  each  body  around  any  point, 
fixed  with  reference  to  tlu>  centre  of  gravity  of  the  system,  by  the  mass  of  the  body. 

This  theorem  is  demonstrated  as  follows :     The  sum 


I 


2  w,(av/,  — .r'i2/i) 

1=0 

is  known  to  be  a  constant  by  t!ie  principle  of  conservation  of  areas.  From  the  ex- 
j)ressi()ii  ({))  for  :»•,,  and  the  corresponding  (expression  for  y„  introducing  the  quantity 
Uoi  as  ill  ((S),  we  have 

(•'i.'/i  •'  illi)  —       *-  „       y.'.j',  k  i^j'.kJt 

J  =0*  ^  0   wr,- 
multiplying  by  «;„  and  then  summing  with  respect  to  /,  we  have 

j  =0  V      0    I  1  -   0       })li       I 

By  the  condition  of  the  orthogonal  system  (S)  the  sum  in  brackets  vanishes  when- 
ever/ is  different  from  /•,  and  beconi(>s  unity  when  these  indices  are  equal.  More- 
over in  (5)  c'y  and    „  vanish  whenever  the  origin  of  co-ordinates  is  fixed  relatively 


30 


G  E  N  K  II  A  L    r  X  T  K  G  H  A  L  S   O  F    PLAN  E  T  A  R  Y   M  O  T I  0  X. 


I  > 


SuLstitutins  for  ^,  ,,  C',  and  ,'  their  oxpressioas  (16),  the  constant  term  of  this 
expression  heconies 

WW. 

But  if  we  a<ld  all  the  values  of  .•  in  (-l),  noting  that  by  the  forui  of  the  general 
integrals  we  have  cv.in.i.u 

h-\-h  +  h  + +  ^,,,=1 

j\+j2+.h  + +;„:,= 0, 


we  find,  also, 
and  hence 


:£/;  =  S'l>/^, 


neon,,,  V.-Thc  constant  part  of  the  living  force,  whicli  is  itself  eqnal  to  the 
constant  //.n  the  integral  of  living  forces,  nsually  expressed  in  the  forin 

IS  represented  hy 

kV,+6,c,  + +&,,„<■,,„), 

as  already  shown  in  § }). 

The  constant  part  of  H  itself  is  therefore  equal  to 

V,+ft,r-.+ +h,„c,„. 

The  equality  of  //  to  the  constant  part  of  T  may  be  shown  by  the  preccdii..-  ^l.eorv 
or  U  may  I.  easily  deduced  directly  trom  the  theorem  of  lini.g  Lr.  as  Jl.own  W 
Jacohi.      (  \or/rsunf/('ii  iihrr  Dumimll,;  p.  •>[)  ) 

The  conditions  that  the  Lagrangiau  coefficients  ,.„/,),  the  sum  of  the-canonical 
auH, lar  velwcitu-s  and  the  .l.tterence  between  ,he  potential  an.l  living  force  are  al 
constant,  give  r,se  to  a  nun.bcn-  of  relations  b..twe.-ii  the  quantities  I  /,  ad  th. 
<U>nvat.ves  w,th  respec.  to  c,  which  I  have  not  jet  t;.„n,l  of  any  use  \n  the  o  e  ^ 
turns  ot  integration.  I  theirtore  omit  to  c.ite  them,  especially  as  tlu-ir  ,.ompl..,e 
expressions  are  rather  complex.  i«<«mpi(i(. 

l^e  i^nns  whi..h  we  have  been  considering  are    those  in  which  it  vould  be 

cce.ss,u>   to  develop  flu.  ..xpress.ons  for  co-ordinates  of  th,.  planets,  if  we  wished 

these  expressions  to  hoi,    true  for  all  tim,>.     The  usual  expivssious  are  su«ici,.utlv 

-•••cct  tor  a  few  centun,-s,  but  fail  .ititvlv  wh,-n  w,-  exten.l  the  tim,-  l.von,!  c-i- 

tmn  bmits      But,  in  the  case  of  th,-  plaiu.fary  system,  we  are  ol.li,...,  ,„  :„„,.„,  , , 

h,.m  tor  th,^  r..as„n  that  formulas  dev..i,.ped  in  multipl,.  of  the  •>;5»  in,I..p,.n.l..„t 

a  guments  o    that  system  w,udd  b,.  uun.anag,.abl,-  iu  p,.a,.ti,.e.      Hut,  i„  ,L  ,.,J 

ot  the  subsuhary  systems,  as  the  Tellurian  ami  Jovian  for  instanc,-,  the  scdar 


wUiLh  UHaiccs  the  ium.b..r  ul  milly  imlqu.n.ient  arpiim.nts  t..  ;)/,-!. 


'.'■.■^.■f" 


a  K  X  10  15  A  L   1  N  T  K  (J  U  A  L  S    O  V    V  L  A  X  K  T  A  K  Y   M  O  T  I  O  N . 


31 


vaiiatious  of  tlic  orbits  are  so  rapid  that  the  apimiximatioii  in  pow(>rs  of  tlu<  tiino 
fails  wen  for  \n-csvn\  uses,  llnicc,  tiu-  lunar  theory,  eonsidered  as  a  problem  of 
three  bodies  oidy,  is  always  treated  in  a  manner  analogous  to  that  in  which  the 
general  theory  of  planetary  motion  has  been  eonsidered  in  the  present  paper,  the 
three  arjrumeiits  introdue(<d  by  the  moon  being  her  mean  longitude,  and  the  longi- 
tudes of  her  node  and  perigee.  In  the  theory  of  Delaunay  the  analo.ry  in  (piestiou 
is  nu)st  easily  seen.  His  A,  ^\  7/,  represent  three  of  our  canonical' elements  e„ 
the  constant  term  of  /<',  to  wliicii  he  constantly  approximates,  is  the  constant  part, 
of  so  much  of  the  expression  for  th(>  living  force  as  contains  /.,  Cr,  and  7/,  by  differ- 
entiating  which  with  respect  to  the  latter  (pmutities,  lie  obtains  the  expressimis  for 
the  motions  of  the  thriM-  arguments. 

Tlie  theory  of  Jupiter's  satellites  has  been  treated  by  ^l.  Souillart  in  such  a 
niam.-r  that  the  <\)-ordinates  may  contain,  instead  of  the  longitudes  of  the  ju-ri- 
ioves,  the  varying  angles  on  which  these  longitudes  de|)end.  His  analytical  theory 
IS  given  in  the  Aiiiialrfi  </r  /"JCcuIr  Xonmtic  Sii/u'rieiirc,  \o\.  '2,  INt!,'). 

It  may  he  hopiMl  that  the  gen(<ral  view  of  the  subject  taken  in  the  presi-nt  paper 
will  afford  a  means  of  introducing  a  more  rigorous  system  of  integration  in  such 
cas(>s.  One  of  the  special  i)rol)Iems  growing  out  of  this  geiu'ral  theorv  will  !)(<  the 
determination  of  the  coeffici(-nts  of  the  time,  A„  /<.,,  etc.,  eitlier  in  terms  of  the  canoni- 
cal constants  c„  c,,  etc.,  m-  of  th(>  largest  of  the  coefJici,-nts  /•.  in  the  expressions  for 
the  co-(mlinafes  of  the  several  planets.  These  coefHcients  are,  approximately,  the 
mean  distances  of  the  planets.  The  (pianlities  />  ought,  perhaj.s,  to  api)ear  as  the 
re.;;s  of  an  e(piation  of  the  '.]u//i  degree,  but  the  writer  h.is  not  yet  succeeded  in 
forn.'iig  any  expression  fitted  to  give  ris.>  to  such  an  cpiation,  excei)t  one  in  which 
only  the  .ijiiares  of  the  (luantities  in  nuestiou  appear. 


ril|)i.,.,aK|l     11V     TIIK    .SMITIISOMAN     1  N  Ml  T  li  T  U)  N  , 

W  A  «  a    1    N  U    I'  1)  \      (■   1  ■!•  V  _ 

l>  K  !■  K  V  M  K  II  .      i  S  7  4  . 


